A physical quantity $$X$$ depends on the velocity of light ($$c$$), Planck's constant ($$h$$), and Newton's gravitational constant ($$G$$) such that the dimensional formula of $$X$$ is identical to that of surface tension. If the relation is expressed as $$X = c^a h^b G^c$$, then find the absolute integer value of $$(a + 2b + 3c)$$.

The dimensional formula of surface tension is  $$[\text{surface\ tension}]=\frac{[\text{force}]}{[\text{length}]}$$

Therefore,  $$[\text{surface\ tension}]=MLT^{-2}L^{-1}=MT^{-2}$$

Given,  $$X=c^ah^bG^c$$

The dimensions of the quantities are

$$[c]=LT^{-1}$$

$$[h]=ML^2T^{-1}$$

$$[G]=M^{-1}L^3T^{-2}$$

Hence,  $$[X]=(LT^{-1})^a(ML^2T^{-1})^b(M^{-1}L^3T^{-2})^c$$

Comparing the powers of $$M,L,T$$ with $$MT^{-2}$$,

$$b-c=1$$

$$a+2b+3c=0$$

$$a+b+2c=2$$

The quantity asked is  $$a+2b+3c$$

From the dimensional equation itself,  $$a+2b+3c=0$$

Hence, the correct answer is 0.